The vectors \(\underset{\sim}{a}, \underset{\sim}{b}\) and \(\underset{\sim}{c}\) have magnitudes 3, 5 and 7 respectively.
 

Given that \(\underset{\sim}{a}+\underset{\sim}{b}+\underset{\sim}{c}=\underset{\sim}{0}\), what is the size of angle \(\theta\) between \(\underset{\sim}{a}\) and \(\underset{\sim}{b}\) ?

1. \(\dfrac{\pi}{6}\)
2. \(\dfrac{\pi}{3}\)
3. \(\dfrac{2 \pi}{3}\)
4. \(\dfrac{5 \pi}{6}\)

For what value of \(m\) is the vector \(\displaystyle \binom{1}{m}\) parallel to the vector \(\displaystyle \binom{2}{6}\)?   (1 mark)

The angle between two unit vectors `underset~a` and `underset~b` is `theta` and  `|underset~a+ underset~b| < 1`.

Which of the following best describes the possible range of values of  `theta` ?

1. `0 <= theta < (pi)/(3)`
2. `0 <= theta < (2pi)/(3)`
3. `(pi)/(3) < theta <= pi`
4. `(2pi)/(3) < theta <= pi`

Consider the vectors  `underset~a = x underset~i + underset~j, \ underset~b = underset~i - underset~j`  and  `underset~c = underset~i + x underset~j`.

Given that  `theta`  is the angle between  `underset~a`  and  `underset~b`,  and  `phi`  is the angle between  `underset~b`  and  `underset~c, cos(theta) cos (phi)`  is

1. `(2(1 + x^2))/(1 - x^2)`
2. `(sqrt2(1 - x^2))/(1 + x^2)`
3. `-((x + 1)^2)/(2(1 + x^2))`
4. `-((x - 1)^2)/(2(1 + x^2))`

For the two vectors  `overset->(OA)`  and  `overset->(OB)`  it is know that

`overset->(OA) · overset->(OB) < 0`

Which of the following statements MUST be true?

1. Either, `overset->(OA)`  is negative and  `overset->(OB)`  is positive, or  `overset->(OA)`  is positive and  `overset->(OB)`  is negative.
2. The angle between  `overset->(OA)`  and  `overset->(OB)`  is obtuse.
3. The product  `|overset->(OA)||overset->(OB)|`  is negative.
4. The points `O`, `A` and `B` are collinear.

Using vectors, calculate the acute angle between the line that passes through  `A(1, 3)`  and  `B(2,–6)`  and the line that passes through  `C(1, 5)`  and  `D(3,–2)`.

Give your answer correct to one decimal place.  (2 marks)

What is the angle between the vectors  `((2),(1))`  and  `((-4),(2))`?

A.   `cos^(-1)(0.06)`

B.   `cos^(-1)(–0.06)`

C.   `cos^(-1)(0.6)`

D.   `cos^(-1)(–0.6)`

If  `theta`  is the angle between  `underset~a = underset~i + 3j`  and  `underset~b = 3underset~i + underset~j`, then find the exact value of  `cos 2theta`.  (2 marks)

Relative to a fixed origin, the points `A`, `B` and `C` are defined respectively by the position vectors  `underset~a = −underset~i - underset~j, \ underset~b = 3underset~i + 2underset~j`  and  `underset~c = −aunderset~i + 2underset~j`, where  `a`  is a real constant.

If the magnitude of angle `ABC`  is  `pi/3`, find `a`.  (3 marks)

Points  `A`  and  `B`  have position vectors  `underset~a = 2underset~i - 2underset~j`  and  `underset~b =  4i + sqrt2 underset~j` respectively.

Find the angle between  `underset~a`  and  `underset~b`  to the nearest minute.  (2 marks)

Consider the vectors given by  `underset ~a = m underset ~i + underset ~j`  and  `underset ~b = underset ~i + m underset ~j`, where  `m in R`.

Find the value(s) of  `m`  if the acute angle between  `underset ~a`  and  `underset ~b`  is 30°.   (2 marks)